Studies over the existence of a certain impulse-based fuzzy integrodifferential equations of the Sobolev type

The present investigation employs impulses and a non-local constraint to prove the existence are some various types of abstract differential and integrodifferential equations related to the Sobolev type. Semigroup theory, specifically variants of constant formula, is utilized to get the analytical results for those equations. Furthermore, findings using the Banach fixed point approach were examined using fuzzy numbers with values spanning the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {E}_n$$\end{document}En range, which includes the normal, convex, upper semi-continuous, and compactly supported interval. A description is given for each situation to illustrate the principle.


Introduction
There are numerous applications for the intriguing theory of differential equations throughout abstract spaces in the fields of analysis and other mathematics.Ordinary differential equations (ODEs), functional differential equations, partial differential equations (PDEs), and sometimes a combination of interacting systems of ordinary and partial differential equations would be used, depending on the nature of the problems.In the fields of applied mathematics, engineering, biology, and the physical sciences, nonlinear differential and integral equations in abstract spaces have been utilized to deal with several problems.With significant applications in many areas of analysis and other disciplines, the theory of nonlinear differential and integral equations in abstract spaces is growing quickly.
The fuzzy semigroups of linear operators to solve fuzzy differential equations were originally proposed in the fuzzy literature by the authors of Gomes et al. [1].Next, fuzzy Cauchy problems were studied by Kaleva [2] using nonlinear iteration semigroups (with exponential formula).Ding and Kandel [3] examined how differential equations and fuzzy sets may be used to create fuzzy logic systems, also known as fuzzy dynamical systems, which are similar to fuzzy neutral functional differential equations.With its wide range of applications, semigroup theory has recently been the subject of much study in the classic literature.For the interested reader, we recommend the fascinating work by Pazy [4], in which the author uses semigroups to solve partial and ordinary differential equations.
PDEs are frequently employed to simulate a broad variety of scientific and engineering issues.They are typically expressed in several forms of differential or integrodifferential equations in abstract spaces.Several writers have studied integrodifferential equations in abstract spaces, including [5][6][7][8].In general, an integrodifferential equation is an abstract representation of a large number of partial integrodifferential equations that emerge in issues involving wave propagation and other physical phenomena.By addressing Sobolev type nonlinear interodifferential equations, Radhakrishnan et al. [9] investigated the existence of Sobolev type nonlinear neutral integrodifferential equations.Many writers have analyzed differential equations of the Sobolev type [10][11][12].
Nonlocal Cauchy problem, namely, the differential equation with a non-local initial condition z(τ 0 ) + e(τ 1 , . . ., τ n , z) = z 0 (0 ≤ τ 0 < τ 1 < . . .< τ n ≤ τ 0 + a and e is a given function) is one of the important topics in the study of analysis.The primary motivation behind interest in the area revolves around the non-local initial condition's high efficacy over the standard one while treating physical issues.In fact, the conventional beginning condition z(0) = z 0 could not be incorporated into a number of fascinating empirical events that the non-local initial condition represents.For instance, the function e(τ 1 , . . ., τ n , z) may be given by (c k , k = 1, . . ., n are constants).In this case, we are per- mitted to have the measurements at τ = 0, τ 1 , . . ., τ n , rather than just at τ = 0.More specially, letting e(τ 1 , . . ., τ n , z) = −z(τ k ) and z 0 = 0 yields a periodic problem and letting e(τ 1 , . . ., τ n , z) = −z(τ 0 ) + z(τ n ) gives a backward problem.Byszewski [13] was the first to delve at the existence of solutions to evolution equations in Banach spaces with non-local constraints.
In reality, experts agree that discretely emerging discontinuities enrich its continuous.The latter are also known as jumps or, from an energy standpoint, impulses.Many evolution processes are accompanied by abrupt shifts in condition at specific points in time.These processes are subject to short-term perturbations that are insignificant in contrast to the process's lifespan.As a result, it's reasonable to presume that these disturbances occur instantly, in the form of impulses.As a result, differential equations with impulsive effects serve as a natural description of observable evolution events in a variety of real-world issues, such as [14][15][16][17][18][19][20][21].In some situations, such as the so-called neutral differential difference equations, the delayed argument occurs in both the derivative of the state variable and the independent variable.A neutral functional differential equation involves the derivatives concerning past events history or derivatives of functionals of the prior history, and the present state of the system.The book on neutral functional differential equations by Hale and Verduyn Lunel [22] and its references are a useful resource.
In accordance with the premise that "all things occurring in the real world are unstable and unexpected, " Zadeh [23] developed fuzzy set theory in 1965.In a number of research areas, the notion was proposed and implemented successfully.This hypothesis has lately been investigated further, with a variety of applications being proposed.To explain fuzzy conceptions, Diamand et al. [24] established the metric space of fuzzy sets theory.Kaleva [25,26] looked at fuzzy differential equations in broad sense.For further discussion on the several types of fuzzy differential equations, see [27][28][29][30][31][32].
Motivated by the literature, we are using the fixed point approach to investigate several types of Sobolev type Fuzzy neutral integro-differential equations with impulses in a fuzzy environment.

Problem formulation
The authors of this paper have to investigate if fuzzy neutral impulsive nonlinear integrodifferential equations of the Sobolev type are subject to non-local conditions where A, B : J → E n denotes a fuzzy coefficient and J = [0, a] .The set of all upper semi continu- ous convex regular numbers that are uncertain, with limited −level intervals is designated as E n .The function F, P , K : J × E n → E n is non- linear fuzzy function and z 0 is a initial value and

Preliminaries
By giving each z ∈ R n a membership grade, a fuzzy sub- set of R n is constructed using a membership function.In this way, the purpose of membership is addressed.
Across the analysis, the subsequent requirements were introduced: w maps R n onto [0, 1], [u] 0 constitutes a finite subset of R n , u is fuzzy convex, and u is upper semi-con- tinuous.In this instance, consider E n represent the space of all fuzzy subsets u of R n , encompassing upper semi- continuous, normal, and fuzzy convex sets along with bounded supports.The space of all fuzzy subsets u of R is precisely expressed by E 1 .
A fuzzy amount A in R comprises a set that is fuzzy defined by the membership function χ A spanning R to [0, 1].A number that is fuzzy A also transforms into with χ A (•) in the closed interval 0 and 1 Let z in R n and D be a nonempty subsets of R n .Now the Hausdroff separation of B from D is defined by Let D and B be nonempty subsets of R n .The Hausdroff separation of B from D is defined by In general, With regard to two mathematical functions A and B , the Hausdroff gap between them can be expressed as and is obviously metric on E n .The writers of this paper make an inference that there is an operator E on E n , which is provided by formula We primary investigate the following fuzzy functional differential equation (FFDE) along with non-local initial condition of Sobolev type where A, B : J → E n denotes a fuzzy coefficient and J = [0, a] , E n is the collection of all upper semi continu- ous convex normal fuzzy numbers with bounded −level intervals.The function F : J × E n → E n is nonlinear fuzzy function and z 0 is a initial value.
is said be a solution, of problem (2.1, 2.2) on J. Remark 2.1 A solution of (2.1, 2.2) satisfies the condition (2.2).Then and Therefore The subsequent assumptions are required in order to demonstrate the existence result: , for ν ∈ J and z ν , y ν ∈ E n .(A 3 ) For our convenience, we choose
Proof Consider the subset X of C(J, E n ) .specified through Again, we define a mapping F : X → X by First to show that the operator F maps X into itself.Now From the assumption Therefore, Hence, Since � < 1 , this show that the operator ζ is contraction on E n and so, by Banach fixed point theorem, there exists a unique fixed point z ∈ F such that F (z(ν)) = z(ν) .This fixed point is the solution of (2.1)-(2.2).Thus Theorem 3.1 is proved.

Case study on Sobolev type of FFDEs
Consider the fuzzy differential equation of the form with a non-local condition ≤�H 1 z(ν), y(ν) . (3.1) The level set of fuzzy number 3 : The -level set of where L f = 4b|x r (ν)) + y r (ν)| meets the inequality stated within the circumstance (A 2 ). where Thus, all conditions of Theorem 3.1 are satisfied.Hence the system (3.1)-(3.2) has a unique fuzzy solution.

Fuzzy neutral integrodifferential equation
We shall explore at the non-linear neutral fuzzy integrodifferential equation of the form in this section.
where A, B : J → E n denotes a fuzzy coefficient and J = [0, a] .The function F, P , K : J × E n → E n is non- linear fuzzy function and z 0 is a initial value.
To prove the existence of (4.1)-(4.2), the following requirements must be fulfilled: (A 4 ) The function P : J × E n → E n satisfy the condition There exists L p ≥ 0. (A 5 ) The function K : J × E n → E n satisfy the condition.
There exists L k ≥ 0.
(A 6 ) For convenience In this section, we'll suppose that there is an operator E on E n , which can be found using the formula with

Definition 4.1 An expression for the integral equation z(ν)
is said to be a fuzzy solution of (4.1, 4.2) on E n .
Remark 4. 1 The fuzzy functional neutral integrodifferential equation (4.1, 4.2) has a fuzzy solution that satisfies (4.3).Therefore, Proof Let F 1 be the subset of C(J, E n ) defined by We define a mapping F : First to show that the operator F maps F 1 into itself.Now

Now,
Therefore, Since § < 1, the above equation, demonstrate the con- traction associated with the operator F over E n and hence by Banach fixed point theorem there exists a unique fixed point z ∈ F 1 such that ( F z)(ν) = z(ν).This fixed point is then the solution of the problem (4.1, 4.2).

Case study: Consider the non-local condition on the fuzzy neutral indegrodifferential equation
The level set of fuzzy number 3 : The -level set of where L f = 4b|z r (ν)) + y r (ν)| satisfies the inequality which is given in condition (A 2 ).
where M c = | n i=1 c i | satisfies the inequality which is given in condition (A 2 ).
Thus, all conditions of Theorem 4.1 are satisfied.Hence the system (4.4,4.5) has a unique fuzzy solution.
Since L B < 1, the above equation, the operator F1 is a contraction on E n , the Banach fixed point theorem indi- cates that there exists a unique fixed point z such that ( F1 z)(ν) = z(ν).At this fixedpoint, the problems (5.1)- (5. 3) arrive at their conclusion.

Case study
Consider the fuzzy neutral indegrodifferential equation with non-local condition

The level set of fuzzy numbers
Now α level set of functions are The -level set of where L f = 3b|x r (τ )) + y r (τ )| complies with the ine- quality indicated within the case of (A 2 ).where M c = | n i=1 c i | complies with the inequality indi- cated within the case of (A 2 ).
where, L i = 3e −2b satisfies the inequality which is given in condition (A 7 ) .As a result, all of Theorem 5.1's require- ments are met.As an outcome, the fuzzy solution for the system (6.1, 6.2, 6.3) appears unique.

Conclusion
The outcomes of this work demonstrated the existence of specific types of impulsive neutral integrodifferential equations with Sobolev-type non-local conditions in a fuzzy environment.Fuzzy intervals that are normal, convex, upper semi-continuous, and compactly supported were used in conjunction with the fixed point strategy to examine the findings.An example is given in order to illustrate the concept for each case.Future research will expand this type of study to include control theory and fractional calculus.